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Theorem dnnumch2 42362
Description: Define an enumeration (weak dominance version) of a set from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Hypotheses
Ref Expression
dnnumch.f 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
dnnumch.a (𝜑𝐴𝑉)
dnnumch.g (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
Assertion
Ref Expression
dnnumch2 (𝜑𝐴 ⊆ ran 𝐹)
Distinct variable groups:   𝑦,𝐹   𝑦,𝐺,𝑧   𝑦,𝐴,𝑧
Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐹(𝑧)   𝑉(𝑦,𝑧)

Proof of Theorem dnnumch2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dnnumch.f . . 3 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
2 dnnumch.a . . 3 (𝜑𝐴𝑉)
3 dnnumch.g . . 3 (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
41, 2, 3dnnumch1 42361 . 2 (𝜑 → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
5 f1ofo 6834 . . . . . 6 ((𝐹𝑥):𝑥1-1-onto𝐴 → (𝐹𝑥):𝑥onto𝐴)
6 forn 6802 . . . . . 6 ((𝐹𝑥):𝑥onto𝐴 → ran (𝐹𝑥) = 𝐴)
75, 6syl 17 . . . . 5 ((𝐹𝑥):𝑥1-1-onto𝐴 → ran (𝐹𝑥) = 𝐴)
8 resss 6000 . . . . . 6 (𝐹𝑥) ⊆ 𝐹
9 rnss 5932 . . . . . 6 ((𝐹𝑥) ⊆ 𝐹 → ran (𝐹𝑥) ⊆ ran 𝐹)
108, 9mp1i 13 . . . . 5 ((𝐹𝑥):𝑥1-1-onto𝐴 → ran (𝐹𝑥) ⊆ ran 𝐹)
117, 10eqsstrrd 4016 . . . 4 ((𝐹𝑥):𝑥1-1-onto𝐴𝐴 ⊆ ran 𝐹)
1211a1i 11 . . 3 (𝜑 → ((𝐹𝑥):𝑥1-1-onto𝐴𝐴 ⊆ ran 𝐹))
1312rexlimdvw 3154 . 2 (𝜑 → (∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴𝐴 ⊆ ran 𝐹))
144, 13mpd 15 1 (𝜑𝐴 ⊆ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wne 2934  wral 3055  wrex 3064  Vcvv 3468  cdif 3940  wss 3943  c0 4317  𝒫 cpw 4597  cmpt 5224  ran crn 5670  cres 5671  Oncon0 6358  ontowfo 6535  1-1-ontowf1o 6536  cfv 6537  recscrecs 8371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372
This theorem is referenced by:  dnnumch3lem  42363  dnnumch3  42364
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